Vector space questions

I'm having trouble with a couple of things written in some notes I'm reading.

Firstly, in stating examples of vector spaces, they sayTrigonometric polynomials - Given n distinct (mod 2π) complex constants λ1,...,λn, the set of all linear combinations of eiλnz forms an n-dimensional complex vector space.

Now does this even make sense? I feel as though it is trying to say that we can have eiλnz for integer λn as the basis for the vector space of functions, i.e like Fourier series, but I'm not sure really.

Secondly, for the complex vector space defined by the vectors in the 2D plane
|r,φ>=(rcosφ,rsinφ)
where vector addition is as usual and scalar multiplication follows the rule
α|r,φ>=||α|r,φ+argα>
I'm having trouble proving that
α(|a>+|b>)=α|a>+α|b>
and
(α+β)|a>=α|a>+β|a>

For the second one, I do
(α+β)|a>=(α+β)|r,φ>
=||(α+β)|r,φ+arg(α+β)>
and I have no idea what to do next, as expanding the modulus/argument doesn't seem possible. Not sure about the second either.

Staff: Mentor

I'm having trouble with a couple of things written in some notes I'm reading.

Firstly, in stating examples of vector spaces, they sayTrigonometric polynomials - Given n distinct (mod 2π) complex constants λ1,...,λn, the set of all linear combinations of eiλnz forms an n-dimensional complex vector space.

Now does this even make sense? I feel as though it is trying to say that we can have eiλnz for integer λn as the basis for the vector space of functions, i.e like Fourier series, but I'm not sure really.

The λi's are complex, not integer.

albega said:

Secondly, for the complex vector space defined by the vectors in the 2D plane
|r,φ>=(rcosφ,rsinφ)
where vector addition is as usual and scalar multiplication follows the rule
α|r,φ>=||α|r,φ+argα>
I'm having trouble proving that
α(|a>+|b>)=α|a>+α|b>
and
(α+β)|a>=α|a>+β|a>

Show us where you've gotten on these. Start with the left side and expand it using the rule above for scalar multiplication.

albega said:

For the second one, I do
(α+β)|a>=(α+β)|r,φ>
=||(α+β)|r,φ+arg(α+β)>
and I have no idea what to do next, as expanding the modulus/argument doesn't seem possible. Not sure about the second either.

For thinking of them as a basis set, first consider what it would take to normalize them.
Then, since the lambdas are distinct, mod 2pi, it can be shown that they are also orthogonal.
This is similar in concept to the fourier basis, but with complex lambdas.
For the last part, you should be able to put it back into complex exponential form and demonstrate the properties you are looking for.

The λi's are complex, not integer.
Show us where you've gotten on these. Start with the left side and expand it using the rule above for scalar multiplication.

Yeah, I know that's what it says, but I just don't understand how that is the case... A quick google of trigonmetric polynomials shows them to be the functions
sin(mx), cos(mx) or exp(imx) for integer m...

For thinking of them as a basis set, first consider what it would take to normalize them.
Then, since the lambdas are distinct, mod 2pi, it can be shown that they are also orthogonal.
This is similar in concept to the fourier basis, but with complex lambdas.
For the last part, you should be able to put it back into complex exponential form and demonstrate the properties you are looking for.

Hmm I think I'm having trouble understanding what it means by mod(2pi), especially with the lambdas being complex. Could you explain please?

Staff: Mentor

Secondly, for the complex vector space defined by the vectors in the 2D plane
|r,φ>=(rcosφ,rsinφ)
where vector addition is as usual and scalar multiplication follows the rule
α|r,φ>=||α|r,φ+argα>
I'm having trouble proving that
α(|a>+|b>)=α|a>+α|b>

It might be easier to start with the right side, α|a>+α|b>.
You could write the vectors a and b as (r1cosφ1) and (r2cosφ2). According to the scalar multiplication rule above, what is αa? What is αb?

Staff: Mentor

Notice that the imaginary part of ##\lambda## will turn into your real part in the exponential, so the mod 2pi only needs to apply to the real part of ##\lambda## to prevent duplication in the imaginary part, since sin(x)=sin(x+2pi).

Notice that the imaginary part of ##\lambda## will turn into your real part in the exponential, so the mod 2pi only needs to apply to the real part of ##\lambda## to prevent duplication in the imaginary part, since sin(x)=sin(x+2pi).

Ok, so are these actually trigonometric polynomials - wikipedia seems to claim they are just sin(mx), cos(mx), exp(imx) for integer m.

I'm confused about the whole mod(2pi) thing - why don't we just say we have all these lambdas and they must be between zero and 2pi?

So the vector space produced by linear combinations of these will be any function?

I'm confused about the whole mod(2pi) thing - why don't we just say we have all these lambdas and they must be between zero and 2pi?

So the vector space produced by linear combinations of these will be any function?

Since pi is irrational the the set of integers times x is equivalent to the set of distinct points in the range 0 to 2pi.
In any case, whether you use mx or lambda x, you end up with the same vector space.